Đáp án:

\(\begin{array}{l}
2,\\
a = 45^\circ \\
3,\\
\left[ \begin{array}{l}
\sin a = \dfrac{{24}}{{25}};\,\,\cos a = \dfrac{7}{{25}}\\
\sin a =  - \dfrac{7}{{25}};\,\,\cos a =  - \dfrac{{24}}{{25}}
\end{array} \right.
\end{array}\)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
2,\\
0 < a < 90^\circ  \Rightarrow \left\{ \begin{array}{l}
0 < \sin a < 1\\
0 < \cos a < 1
\end{array} \right.\\
{\left( {\sin a + \cos a} \right)^2} = {\sin ^2}a + 2\sin a.\cos a + {\cos ^2}a\\
 = \left( {{{\sin }^2}a + {{\cos }^2}a} \right) + 2\sin a.\cos a = 1 + 2.\dfrac{1}{2} = 1 + 1 = 2\\
\left\{ \begin{array}{l}
0 < \sin a < 1\\
0 < \cos a < 1
\end{array} \right. \Rightarrow 0 < \sin a + \cos a < 2\\
 \Rightarrow \sin a + \cos a = \sqrt 2 \\
 \Leftrightarrow \sin a = \sqrt 2  - \cos a\\
\sin a.\cos a = \dfrac{1}{2}\\
 \Leftrightarrow \left( {\sqrt 2  - \cos a} \right).\cos a = \dfrac{1}{2}\\
 \Leftrightarrow \sqrt 2 \cos a - {\cos ^2}a = \dfrac{1}{2}\\
 \Leftrightarrow 2\sqrt 2 \cos a - 2{\cos ^2}a = 1\\
 \Leftrightarrow 2{\cos ^2}a - 2\sqrt 2 \cos a + 1 = 0\\
 \Leftrightarrow {\left( {\sqrt 2 \cos a} \right)^2} - 2.\sqrt 2 \cos a.1 + {1^2} = 0\\
 \Leftrightarrow {\left( {\sqrt 2 \cos a - 1} \right)^2} = 0\\
 \Leftrightarrow \sqrt 2 \cos a - 1 = 0\\
 \Leftrightarrow \cos a = \dfrac{1}{{\sqrt 2 }}\\
0 < a < 90^\circ  \Rightarrow a = 45^\circ \\
3,\\
\sin a - \cos a = \dfrac{{17}}{{25}}\\
 \Leftrightarrow \sin a = \cos a + \dfrac{{17}}{{25}}\\
{\sin ^2}a + {\cos ^2}a = 1\\
 \Leftrightarrow {\left( {\cos a + \dfrac{{17}}{{25}}} \right)^2} + {\cos ^2}a = 1\\
 \Leftrightarrow {\cos ^2}a + 2.\cos a.\dfrac{{17}}{{25}} + {\left( {\dfrac{{17}}{{25}}} \right)^2} + {\cos ^2}a = 1\\
 \Leftrightarrow 2{\cos ^2}a + \cos a.\dfrac{{34}}{{25}} + \dfrac{{289}}{{625}} = 1\\
 \Leftrightarrow 2{\cos ^2}a + \dfrac{{34}}{{25}}\cos a - \dfrac{{336}}{{625}} = 0\\
 \Leftrightarrow {\cos ^2}a + \dfrac{{17}}{{25}}\cos a - \dfrac{{168}}{{625}} = 0\\
 \Leftrightarrow \left( {{{\cos }^2}a - \dfrac{7}{{25}}\cos a} \right) + \left( {\dfrac{{24}}{{25}}.\cos a - \dfrac{{168}}{{625}}} \right) = 0\\
 \Leftrightarrow \cos a.\left( {\cos a - \dfrac{7}{{25}}} \right) + \dfrac{{24}}{{25}}.\left( {\cos a - \dfrac{7}{{25}}} \right) = 0\\
 \Leftrightarrow \left( {\cos a - \dfrac{7}{{25}}} \right)\left( {\cos a + \dfrac{{24}}{{25}}} \right) = 0\\
 \Leftrightarrow \left[ \begin{array}{l}
\cos a - \dfrac{7}{{25}} = 0\\
\cos a + \dfrac{{24}}{{25}} = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\cos a = \dfrac{7}{{25}}\\
\cos a =  - \dfrac{{24}}{{25}}
\end{array} \right.\\
\sin a = \cos a + \dfrac{{17}}{{25}} \Rightarrow \left[ \begin{array}{l}
\sin a = \dfrac{{24}}{{25}};\,\,\cos a = \dfrac{7}{{25}}\\
\sin a =  - \dfrac{7}{{25}};\,\,\cos a =  - \dfrac{{24}}{{25}}
\end{array} \right.
\end{array}\)